Calculus/Inverting vector calculus operators

In the chapter on vector calculus, the differential operator of the gradient (∇f{\displaystyle \nabla f}), the divergence (∇⋅F{\displaystyle \nabla \cdot \mathbf {F} }), and the curl (∇×F{\displaystyle \nabla \times \mathbf {F} }) were introduced. This chapter will focus on inverting these differential operators.

Consider the piece-wise function f:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by: f(x)={−1−12x(x<0)3−x(x≥0){\displaystyle f(x)=\left\{{\begin{array}{cc}-1-{\frac {1}{2}}x&(x<0)\\3-x&(x\geq 0)\end{array}}\right.}. It is common to accept that f(x){\displaystyle f(x)} is not differentiable at x=0{\displaystyle x=0} and that f′(x)={−12(x<0)−1(x>0){\displaystyle f'(x)=\left\{{\begin{array}{cc}-{\frac {1}{2}}&(x<0)\\-1&(x>0)\end{array}}\right.}. With the Dirac delta function, the derivative of f{\displaystyle f} can be expressed as f′(x)=4δ(x)+{−12(x<0)0(x=0)−1(x>0){\displaystyle f'(x)=4\delta (x)+\left\{{\begin{array}{cc}-{\frac {1}{2}}&(x<0)\\0&(x=0)\\-1&(x>0)\end{array}}\right.}. With this derivative, part II of the fundamental theorem of calculus holds even for intervals that contain x=0{\displaystyle x=0}.

In this chapter, it will generally be assumed that all scalar fields f:R3→R{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} } and vector fields F:R3→R3{\displaystyle \mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}} are continuous and differentiable everywhere. However if this is not the case, the Dirac delta function will be used to model the derivative operators at points of non-differentiability.

Given a collection C1,C2,…,Ck{\displaystyle C_{1},C_{2},\dots ,C_{k}} of oriented paths, then the vector field δ→(q;C1)+δ→(q;C2)+⋯+δ→(q;Ck){\displaystyle {\vec {\delta }}(\mathbf {q} ;C_{1})+{\vec {\delta }}(\mathbf {q} ;C_{2})+\dots +{\vec {\delta }}(\mathbf {q} ;C_{k})} effectively denotes the "superposition" of C1,C2,…,Ck{\displaystyle C_{1},C_{2},\dots ,C_{k}}. This superposition is referred to as a "multi-path". Not all paths have to have a weight of 1. With multi-path 12δ→(q;C1)+12δ→(q;C2){\displaystyle {\frac {1}{2}}{\vec {\delta }}(\mathbf {q} ;C_{1})+{\frac {1}{2}}{\vec {\delta }}(\mathbf {q} ;C_{2})}, the weights on C1{\displaystyle C_{1}} and C2{\displaystyle C_{2}} are both 0.5{\displaystyle 0.5}. This multi-path is an even 50%/50% superposition between C1{\displaystyle C_{1}} and C2{\displaystyle C_{2}}.

Any vector field F{\displaystyle \mathbf {F} } can be envisioned as a superposition of a possibly infinite number of paths. Each path may have an infinitesimal weight. When a vector field is envisioned as a multi-path, the decomposition into individual paths is not unique. When vector field F{\displaystyle \mathbf {F} } denotes a multi-path, ∇⋅F{\displaystyle \nabla \cdot \mathbf {F} } is the net density of path origin points minus the density of destination points: the path starting point density.

When ∇⋅F=0{\displaystyle \nabla \cdot \mathbf {F} =0} everywhere, F{\displaystyle \mathbf {F} } can be envisioned as a superposition of a possibly infinite number of paths that are either closed or extend to infinity. If it is also the case that F(q){\displaystyle \mathbf {F} (\mathbf {q} )} is O(1/|q|αF){\displaystyle O(1/|\mathbf {q} |^{\alpha _{F}})} for some αF>2{\displaystyle \alpha _{F}>2}, then all of the paths have to close and F{\displaystyle \mathbf {F} } is effectively a "multi-loop". (F(q){\displaystyle \mathbf {F} (\mathbf {q} )} is O(1/|q|αF){\displaystyle O(1/|\mathbf {q} |^{\alpha _{F}})} if and only if there exists some threshold c1>0{\displaystyle c_{1}>0} and factor c2>0{\displaystyle c_{2}>0} such that ∀q∈R3:|q|>c1⟹|F(q)|<c2(1/|q|αF){\displaystyle \forall \mathbf {q} \in \mathbb {R} ^{3}:|\mathbf {q} |>c_{1}\implies |\mathbf {F} (\mathbf {q} )|<c_{2}(1/|\mathbf {q} |^{\alpha _{F}})})

A demonstration of how a divergence free vector field can be envisioned as the superposition of several simple loops.

In the image to the right, a divergence free vector field that denotes flow density is decomposed into the superposition of multiple simple loops. 2 dimensional space is depicted as a lattice of infinitely small squares. The vector field is the top-left section. The flow along each horizontal edge, denoted by the direction and number of arrows, is the horizontal component of the vector field at the current edge (or neighboring vertex). The flow along each vertical edge, denoted by the direction and number of arrows, is the vertical component of the vector field at the current edge (or neighboring vertex). The remaining 3 sections show 3 simple loops whose superposition forms the vector field.

Any vector field F{\displaystyle \mathbf {F} } can be envisioned as a superposition of a possibly infinite number of surfaces. Each surface may have an infinitesimal weight. When a vector field is envisioned as a multi-surface, the decomposition into individual surfaces is not unique. When vector field F{\displaystyle \mathbf {F} } denotes a multi-surface, ∇×F{\displaystyle \nabla \times \mathbf {F} } is the multi-loop that is the counter-clockwise oriented boundary of the multi-surface denoted by F{\displaystyle \mathbf {F} }.

When ∇×F=0{\displaystyle \nabla \times \mathbf {F} =\mathbf {0} } everywhere, F{\displaystyle \mathbf {F} } can be envisioned as a superposition of a possibly infinite number of surfaces that are either closed with no boundaries or extend to infinity. If it is also the case that F(q){\displaystyle \mathbf {F} (\mathbf {q} )} is O(1/|q|αF){\displaystyle O(1/|\mathbf {q} |^{\alpha _{F}})} for some αF>1{\displaystyle \alpha _{F}>1}, then all of the surfaces have to close without extending to infinity and F{\displaystyle \mathbf {F} } is effectively a "multi closed surface".

Given a curve C{\displaystyle C} and a surface σ{\displaystyle \sigma }, the net number of times C{\displaystyle C} passes through σ{\displaystyle \sigma } in the preferred direction, denoted by N{\displaystyle N}, is given by:

An example application of multi-paths and multi-surfaces is given in the box below:

Electromagnetic Induction

Given a closed loop C{\displaystyle C} that is carrying an electric current of I{\displaystyle I}, the magnetic field B{\displaystyle \mathbf {B} } generated by this closed loop obeys the following equations:

Let C1{\displaystyle C_{1}} and C2{\displaystyle C_{2}} be two closed loops which are the counterclockwise boundaries of surfaces σ1{\displaystyle \sigma _{1}} and σ2{\displaystyle \sigma _{2}}. Let B1{\displaystyle \mathbf {B} _{1}} denote the magnetic field generated by running an electric current of I1{\displaystyle I_{1}} around C1{\displaystyle C_{1}}. Note that B1{\displaystyle \mathbf {B} _{1}} is proportional to I1{\displaystyle I_{1}}. Now let ΦB,2{\displaystyle \Phi _{B,2}} denote the total flux of B1{\displaystyle \mathbf {B} _{1}} through σ2{\displaystyle \sigma _{2}} in the preferred direction: ΦB,2=∬q∈σ2B1(q)⋅dS{\displaystyle \Phi _{B,2}=\iint _{\mathbf {q} \in \sigma _{2}}\mathbf {B} _{1}(\mathbf {q} )\cdot \mathbf {dS} }. Note that since ∇⋅B1=0{\displaystyle \nabla \cdot \mathbf {B} _{1}=0}, that ΦB,2{\displaystyle \Phi _{B,2}} is a function of C2{\displaystyle C_{2}} instead of σ2{\displaystyle \sigma _{2}}. It is the case that ΦB,2{\displaystyle \Phi _{B,2}} is proportional to I1{\displaystyle I_{1}}, and the constant of proportionality M(C1,C2)=ΦB,2I1{\displaystyle M(C_{1},C_{2})={\frac {\Phi _{B,2}}{I_{1}}}} is the "mutual inductance" from C1{\displaystyle C_{1}} to C2{\displaystyle C_{2}}.

The mutual inductance M(C1,C2){\displaystyle M(C_{1},C_{2})} is purely a function of C1{\displaystyle C_{1}} and C2{\displaystyle C_{2}}. It is also the case that the mutual inductance is symmetric: M(C1,C2)=M(C2,C1){\displaystyle M(C_{1},C_{2})=M(C_{2},C_{1})}. The symmetry of the mutual inductance is not obvious, and while the symmetry is apparent from explicit formulas for the mutual inductance such as the "Neumann formula" [1], the symmetry can be made clear by interpreting the magnetic field as both a multi-loop and a multi-surface.

The vector field b1=B1μ0I1{\displaystyle \mathbf {b} _{1}={\frac {\mathbf {B} _{1}}{\mu _{0}I_{1}}}} satisfies both ∇⋅b1=0{\displaystyle \nabla \cdot \mathbf {b} _{1}=0} and ∇×b1=δ→(q;C1){\displaystyle \nabla \times \mathbf {b} _{1}={\vec {\delta }}(\mathbf {q} ;C_{1})}. While not proven here, the magnetic field generated by a closed loop of current that does not extend to infinity is O(1/|q|3){\displaystyle O(1/|\mathbf {q} |^{3})} so b1{\displaystyle \mathbf {b} _{1}} is both a multi-loop and a multi-surface with counterclockwise boundary C1{\displaystyle C_{1}}. Doing the same for loop C2{\displaystyle C_{2}} gives b2=B2μ0I2{\displaystyle \mathbf {b} _{2}={\frac {\mathbf {B} _{2}}{\mu _{0}I_{2}}}} where B2{\displaystyle \mathbf {B} _{2}} is the magnetic field generated solely by running a current of I2{\displaystyle I_{2}} around C2{\displaystyle C_{2}}. b2{\displaystyle \mathbf {b} _{2}} is both a multi-loop and a multi-surface with counterclockwise boundary C2{\displaystyle C_{2}}.

To analyse whether or not the integral diverges due to the pole/singularity or infinite range, the volume integral will be expressed as the integral of concentric spherical shells centered on q{\displaystyle \mathbf {q} }:

The inner surface integral does not present any irregularities. The lower bound of r=0{\displaystyle r=0} denotes the pole, and the upper bound of r=+∞{\displaystyle r=+\infty } denotes the infinite range.

This property implies that given a sufficiently large R≫0{\displaystyle R\gg 0}, that any path integral between any two points outside of the sphere B(0;R){\displaystyle B(\mathbf {0} ;R)} that remains outside of the sphere is arbitrarily small. Therefore the origin point q0{\displaystyle \mathbf {q} _{0}} can freely shift between points at infinity.

This new formula f(q)=14π∭q′∈R3F(q′)⋅(q−q′)|q−q′|3dV′{\displaystyle f(\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'} is a volume integral that expresses the potential f{\displaystyle f} as a linear combination of functions that exhibit a degree of spherical symmetry. This formula is similar to the inverse square law for the inverse of the divergence.

Next, the above formula will be derived using a Green's function approach in a manner similar to the inverse square law for the divergence operator.

This section will derive a formula identical to the formula above using a Green's function approach. While the derivation will be complicated and the result will not be new, the derivation itself will yield many interesting intermediate results. Again, it will be assumed that F(q){\displaystyle \mathbf {F} (\mathbf {q} )} is O(1/|q|αF){\displaystyle O(1/|\mathbf {q} |^{\alpha _{F}})} as |q|→+∞{\displaystyle |\mathbf {q} |\to +\infty } for some αF>1{\displaystyle \alpha _{F}>1}. It will also be assumed that F{\displaystyle \mathbf {F} } is continuous.